#8197584

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4967 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(11,15)(13,16)(17,18,23,24)(19,21,25,20)(22,26,31,32)(27,29,30,28);(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(18,24)(20,21)(26,32)(28,29);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,20)(21,24)(22,30)(23,25)(26,29)(27,31)(28,32);(9,10)(11,13)(12,14)(15,16)(17,20,23,21)(18,25,24,19)(22,28,31,29)(26,30,32,27);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,15)(14,16)(17,21,23,20)(18,19,24,25)(22,29,31,28)(26,27,32,30);(1,2)(3,6)(4,5)(7,8)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,32)(25,30)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104780);